#6833993

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1202 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2,4,5)(3,6,7,8)(9,10,14,17)(11,18,21,15)(12,16,13,22)(19,24,20,23)(27,34)(33,39)(35,38)(37,40);(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,30)(27,35)(28,31)(32,36)(33,39)(34,38);(2,5)(6,8)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,26,31,36,29,32,28,30)(27,33,38,40,34,39,35,37);(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,32)(30,36)(33,39)(37,40);(1,3,4,7)(2,6,5,8)(11,21)(15,18)(19,20)(23,24)(25,27,28,35,29,34,31,38)(26,33,30,37,32,39,36,40)

Polycyclic presentation:

2^9|a^2=fg,c^2=fhi,d^2=f,e^2=gh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6741189);